How do you find the discriminant and how many solutions does n^2-3n-40=0 have?

1 Answer
Tony B
May 3, 2018

Given: n^2-3n-40=0

color(blue)("General observation (reading the equation)")

As the n^2 term is positive the graph is of generic shape uu

The n term is negative so the the vertex is to the right of the y-axis thus n_("vertex")>0

As the vertex is to the right of the y-intercept then the y_("vertex") < y_("intercept") and the graph 'crosses the n-axis.

Thus there are two solutions to n^2-3n-40=0
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Deriving the determinant and thus the count of solutions")

Consider the standard form 0=y=ax^2+bx+c

where x=(-b+-sqrt(b^2-4ac))/(2a)

The determinant part is b^2-4ac

In this case: a=+1; b=-3 and c=-40 giving:

b^2-4ac ->(-3)^2-4(1)(-40) = 169

As the determinant is greater than 0 it also tells us that
color(brown)(ul("there are 2 solutions."))

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